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This question is only about terminology. Inside a category we have the standard wordings:

  1. An arrow $f: X \rightarrow Y$ is an isomorphism if there is another arrow $g: Y \rightarrow X$ such that $g \circ f = id_X$ [added] and $f \circ g = id_Y$.

  2. Two objects $X,Y$ are isomorphic if there is an isomorphism $f: X \rightarrow Y$.

I wonder whether there are comparably catchy names for the corresponding functorial cases. All I could find so far is:

  • A functor $F : C \rightarrow D$ yields an equivalence of categories if there is another functor $G : D \rightarrow C$ such that $G \circ F \simeq \mathsf{I}_C$ and $F \circ G \simeq \mathsf{I}_D$ (plus further conditions)

I already find this terminology rather clumsy, but I did not find at all a phrase that could replace $\dots$ in the following definition:

  • Two objects $X\in C,Y \in D$ are $\dots$ if there is an "equivalence functor" $F: C \rightarrow D$ such that $F(X) = Y$.

Did I just miss something?

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    @Hans: To consider an example I know well: the equivalence between crossed modules $\mathcal M$ and group objects $\mathcal G$ in the category of groupoids. One is interested in which specific structures of one kind correspond to those of the other, and one would probably say this particular crossed module is _equivalent_ to this particular group groupoid (or an isomorphic copy, of course). But one wants to keep _isomorphism_ for its standard use.2012-11-19

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Well, it is not usual to relate two objects from different categories. I would say, that $Y$ is the 'correspondent' of $X$, or something like this..

However, one can still say that $X$ and $Y$ are isomorphic, due to the following fact:

Categories $C$ and $D$ are equivalent if and only if there is a category $E$ and full embeddings $\phi:C\to E$ and $\psi:D\to E$ such that for all $X\in C$ there is an $Y\in D$ 'isomorphic to $X$ in $E$', meaning that $\phi(X)\cong \psi(Y)$ in $E$, and for all $Y\in D$ there is an $X\in C$ isomorphic to $Y$ in $E$.

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    @Omar: You can say "F(X) corresponds to$X$under F" for *every* mapping F. You will say "F(X) is isomorphic to X under F" only for special mappings$F$(e.g. isomorphisms). Thus, you have defined "is isomorphic" first. In this vein you might define "F(X) ... to X under F" for even other special mappings F (e.g. equivalence functors).2012-11-13