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Katsuki materializes on the surface realm, his daughter swaddled and tightly wrapped in his arms to protect her from the winter chill. Inko still hasn't let up, despite it officially being spring, and won't for another month, still angry after all these years that she has to abide by this agreement. It's not like Katsuki's all that happy either, but he deals with it.
For Izuku.
It's not a long walk through the snowy forest, and soon enough he's able to unswaddle his child, freeing her struggling arms so she can play with the gems that had been sewn into his clothes. It's always warm near the little cottage, most of the land used for the orchards and gardens, warm enough that even the persistent winter's chill can't touch them here.
Seeing Izuku setting out plates filled with the fruits of his labor and cups of ambrosia warms his heart just as much as the heat, melting away the ice that had formed from all the nights in the dark with a cold bed and no Queen by his side.
"There's my little princess," Izuku coos, taking their daughter from Katsuki's arms; letting the blanket fall free to show the impressive dress the shades had dressed her in for their outing. None who looked upon her would be able to deny she was royal, and none who wished to remain alive would be able to deny she was a god. Not that it mattered to Izuku, who just missed holding her in his arms, missed having access to his little girl every second of the day.
"And there's my broody husband," Izuku teased, black vines growing from the ground to lift him to Katsuki's height, their lips perfectly aligned in their kiss.
In another time, he'd beg for more, the only one to whom the God of the Underworld would bend his knee. But for now, Katsuki accepted this single sip and stored it away, cursing Mina for blessing and cursing him with these feelings, this affection, damn goddess of love.
"Come," Izuku said with a smile, vine gone, feet upon the ground. Their daughter on his hip, his hand extended out towards Katsuki, "and tell me about home."
Katsuki takes it and lets his husband lead him toward the little table Izuku had set up, where they sit, eat, and talk.
Where Izuku talks about the visits he's had with his mother and the other Olympians, how they've asked about Katsuki, encouraging them all to come to their next family get-together, when their daughter is older.
Where Katsuki tells Izuku about their realm, about the shades and their friends, about the servants and their dog. About things going well and troubles they'll need to take care of when Izuku returns, his own powers needed as well as Katsuki's to keep their world in order - he was Izuku, yes, the God of Spring, but he was also Deku, the God of Ghosts and Destruction, and was just as needed as Katsuki was himself.
Together they sat and ate and talked, their family complete, under the vast sky and warm sun so far from the realm they ruled.
A peaceful existence, until she began to cry.
The agreement they'd made all those years ago to keep Inko from setting the world on ice was more of a curse now, months of distance and travel for the sake of humanity instead of just the natural rhyme their days had fallen into. It'd still hurt before when Izuku had had to leave, when their time together was measured in seasonal visits instead of lives lived together, but now it was painful for their family to be incomplete.
But a few months to keep humanity alive was worth it, according to Izuku. Katsuki wasn't so sure, but had to consent.
Still, even though they could visit, their daughter was still young. Unused to the mortal realm, unused to the pull of the Styx trying to call her home, at its mercy for however long it let her wander from its banks. In a few years, she'd be able to withstand the longer visits to the upper realm. Would be able to fight against the painful pull, and no matter the season most of their days would be together.
For now, though, their visits were short, her little body relying on the world in which she'd been born.
They both knew what her cries meant.
A kiss on her forehead, a perhaps too-tight hug, and Izuku handed her back to Katsuki, the smile on his face forced so their parting wouldn't be tearful again.
Another kiss between them - a second, something they needed, their 'I love you's, 'I need you's, 'I miss you's in that kiss because the words had been said too many times before to say them again - before Katsuki pulled back, giving the space he needed for his powers to work.
"I'll be back, Izuku."
"I know, Kacchan. I'll see you tomorrow."
It wasn't painful, letting the Styx take them. It was almost soothing, giving in to the power that pulled upon their hearts, their souls, their very beings, stopping his daughter's cries as they were tugged back home. As he walked from the pool of blood that marked the entrance of The House, their home, where servants were already waiting with towels, papers, and toys for the little one.
Just like they'd be tomorrow, and the next day, and the next, until the warm months were over, and Izuku relaxed into the Styx as well and came home with them.
A day that couldn't come soon enough.