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Suppose, $f: a \longrightarrow b$ and $g: c \longrightarrow d$ are arrows of category $C$ such that $\exists $ isomorphisms $h:a \longrightarrow c$ and $k: b \longrightarrow d$ which satisfy : $k \circ f = g \circ h$. Conjugated arrows?

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$g \circ h : a \longrightarrow d$, not the other way –  alancalvitti Jul 20 '12 at 17:51
You could say they are isomorphic in the functor category $[\mathbb{2}, \mathcal{C}]$, where $\mathbb{2}$ is the finite category $\{ \bullet \to \bullet \}$. –  Zhen Lin Jul 20 '12 at 17:58
@Zhen: also known as the arrow category over $\mathcal C$. –  Henning Makholm Jul 20 '12 at 19:08
I'd just call them isomorphic (it should generally be obvious that it is with respect to the relevant arrow category). –  Tilo Wiklund Jul 21 '12 at 0:24
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