Let $\mathcal{C}$ and $\mathcal{D}$ be categories, $F:\mathcal{C}\to\mathcal{D}$ be a (covariant or contravariant) functor and let $\alpha\in\mathrm{Hom}(A,B)$ be a morphism, where $A,B\in\mathrm{Ob}(\mathcal{C})$

Definition. A morphism $\alpha:A\to B$ is an $F$ textit-isomorphism if there exists a morphism $\beta:B\to A$ such that $F(\beta\circ\alpha)=\mathrm{id}_{F(A)}$ and $F(\alpha\circ\beta)=\mathrm{id}_{F(B)}$ (for contravariant functors reverse the order of composition).

Note that each isomorphism in $\mathcal{C}$ is an $F$ isomorphism for each functor $F$ The converse is also true, i.e. if $\alpha$ is an $F$ isomorphism for each functor $F$ then $\alpha$ is an isomorphism. On the other hand there are $F$ isomorphisms which are not isomorphisms.

Example. $1)$ Let $X$ be an object in $\mathcal{D}$ and define $F_{X}:\mathcal{C}\to\mathcal{D}$ as follows: for $A\in\mathrm{Ob}(\mathcal{C})$ put $F_{X}(A)=X$ and for $\alpha\in\mathrm{Hom}(A,B)$ put $F_{X}(\alpha)=\mathrm{id}_{X}$ This is the constant functor and if $\alpha\in\mathrm{Hom}(A,B)$ and $\mathrm{Hom}(B,A)\neq\emptyset$ then $\alpha$ is an $F_{X}$ isomorphism (although it does not have to be an isomorphism).

$2)$ Let $\mathcal{T}\mathrm{op}^{*}$ be the category of pointed topological spaces and continous maps preserving based point, $\mathcal{S}\mathrm{et}$ be the category of sets and functions, $\mathcal{G}\mathrm{r}$ be the category of groups and homomorphisms. Consider the functor $\pi:\mathcal{T}\mathrm{op}^{*}\to\mathcal{S}\mathrm{et}\times\mathcal{G}\mathrm{r}\times\mathcal{G}\mathrm{r}\times\cdots$ defined by: $$\pi(X,x_{0})=(\pi_{0}(X,x_{0}),\pi_{1}(X,x_{0}),\pi_{2}(X,x_{0}),\pi_{3}(X,x_{0}),\ldots);$$ $$\pi(f)=(\pi_{0}(f),\pi_{1}(f),\pi_{2}(f),\pi_{3}(f),\ldots),$$ where $\pi_{n}$ is the $n$ th homotopy group functor. Then $\pi$ isomorphism is a weak homotopy equivalence and it is known (due to Whitehead) that each weak homotopy equivalence between pointed $\mathrm{CW}$ complexes is the homotopy equivalance. On the other hand there are weak homotopy equivalences which are not homotopy equivalences.