This question is only about terminology. Inside a category we have the standard wordings:
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 $f \circ g = id_Y$.
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?