Recall that the simplex category $\Delta$ is dual to the category of intervals $\mathbb{I}$. By $\Delta$ I mean the category of finite ordinals $\mathbf{n} \in \omega$ with monotone functions between them, and by $\mathbb{I}$ I mean the category of positive finite ordinals with monotone functions that preserve the maximum and minimum elements. The duality is given by a contravariant functor $F : \Delta^{\operatorname{op}} \to \mathbb{I} $, $F(\mathbf{n}) = \mathbf{n+1}$, and if $f \in \operatorname{Hom}_\Delta(\mathbf{n}, \mathbf{m})$ then $$F(f) = i \mapsto \min(\{j\in\mathbf{n} \mid i\leq f(j)\}\}\cup\{\mathbf{n}\})$$ In particular, $F$ gives a bijection between monomorphisms $\mathbf{n} \to \mathbf{m}$ in $\Delta$ and epimorphisms $\mathbf{m+1}\to \mathbf{n+1}$ in $\mathbb{I}$. But actually, these are the same as just monotone injections $\mathbf{n} \to \mathbf{m}$ and monotone surjections $\mathbf{m+1}\to \mathbf{n+1}$, the latter being also epimorphisms $\mathbf{m+1}\to \mathbf{n+1}$ in $\Delta$. So this gives a bijection between monomorphisms $\mathbf{n} \to \mathbf{m}$ in $\Delta$ and epimorphisms $\mathbf{m+1}\to \mathbf{n+1}$ in $\Delta$, and it's not hard to see its composition-reversing. Seems reasonable to call it a transpose?
I haven't seen this idea written down anywhere, and it's very possible I'm overcomplicating things. Probably you can see this a lot more combinatorially. Has anyone seen this before, and if so what is it called/where did you see it?