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Let $f: x\to y$ and $f':x' \to y'$ be arrows in a category ${\bf C}$, where $x\cong x'$ and $y\cong y'$. Assume that for all isomorphisms $v: x \to x'$ and $u: y\to y'$, the corresponding diagram commutes, ie., $f'\cdot v = u \cdot f$. Is there a name for such (seemingly closely related) arrows?

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