2 Texified, since question was on front-page anyway

One general rule that unites some of the examples above is that if you have two categories whose objects are sets endowed with some structure, and there is an equivalence between these two categories that assigns to a set with a structure the same set with a different (but equivalent) structure, than such an equivalence of categories is an isomorphism of categories. One can also have objects of some other fixed category in place of sets and some collections of morphisms in place of the structures on sets (see the very last example below).

To give a simple nontrivial example of this, the category of G-modules $G$-modules for a group G $G$ is isomorphic to the category of modules over the group ring Z[G], $\mathbb{Z}[G]$, or the category of modules over a Lie algebra g $\mathfrak{g}$ is isomorphic to the category of modules over its enveloping algebra U(g), $U(\mathfrak{g})$, or the category of comodules over a finite-dimensional coalgebra C $C$ is isomorphic to the category of modules over the dual algebra C*.$C^\ast$.

Another series of examples of isomorphisms of categories is provided by equivalences between categories whose classes of objects are the same though morphisms are different but isomorphic. This includes equivalences between various quotient categories or localizations of a given category (which all have the same objects as the original category).

Here is another example of this kind. Let C $C$ be a category, R:C->C $R:C\rightarrow C$ be a monad on C, $C$, and L:C->C $L:C\rightarrow C$ be a functor left adjoint to C. $C$. Then L $L$ is a comonad. The categories of R-algebras $R$-algebras and L-coalgebras $L$-coalgebras in C $C$ can be quite different. However, one can consider the category of free R-algebras $R$-algebras in C; $C$; this is a category whose objects are formally just the objects X $X$ of C $C$ while morphisms X->Y $X\rightarrow Y$ are the R-algebra $R$-algebra morphisms R(X)->R(Y)$R(X)\rightarrow R(Y)$. Analogously one defines the category of cofree L-coalgebras $L$-coalgebras in C $C$ whose objects are the objects X $X$ of C $C$ and morphisms are the L-coalgebra $L$-coalgebra morphisms L(X)->L(Y)$L(X)\rightarrow L(Y)$. Then the categories of free R-algebras $R$-algebras and cofree L-coalgebras $L$-coalgebras are isomorphic; this is called the isomorphism of Kleisli categories. To give a concrete example of this, the categories of cofree left comodules and free left contramodules over a given coalgebra are isomorphic.

To compare, when L:C->C $L:C\rightarrow C$ is a monad and R:C->C $R:C\rightarrow C$ is right adjoint to L, $L$, then R $R$ is a comonad and the whole categories of L-algebras $L$-algebras and R-coalgebras $R$-coalgebras in C $C$ are isomorphic.

1

One general rule that unites some of the examples above is that if you have two categories whose objects are sets endowed with some structure, and there is an equivalence between these two categories that assigns to a set with a structure the same set with a different (but equivalent) structure, than such an equivalence of categories is an isomorphism of categories. One can also have objects of some other fixed category in place of sets and some collections of morphisms in place of the structures on sets (see the very last example below).

To give a simple nontrivial example of this, the category of G-modules for a group G is isomorphic to the category of modules over the group ring Z[G], or the category of modules over a Lie algebra g is isomorphic to the category of modules over its enveloping algebra U(g), or the category of comodules over a finite-dimensional coalgebra C is isomorphic to the category of modules over the dual algebra C*.

Another series of examples of isomorphisms of categories is provided by equivalences between categories whose classes of objects are the same though morphisms are different but isomorphic. This includes equivalences between various quotient categories or localizations of a given category (which all have the same objects as the original category).

Here is another example of this kind. Let C be a category, R:C->C be a monad on C, and L:C->C be a functor left adjoint to C. Then L is a comonad. The categories of R-algebras and L-coalgebras in C can be quite different. However, one can consider the category of free R-algebras in C; this is a category whose objects are formally just the objects X of C while morphisms X->Y are the R-algebra morphisms R(X)->R(Y). Analogously one defines the category of cofree L-coalgebras in C whose objects are the objects X of C and morphisms are the L-coalgebra morphisms L(X)->L(Y). Then the categories of free R-algebras and cofree L-coalgebras are isomorphic; this is called the isomorphism of Kleisli categories. To give a concrete example of this, the categories of cofree left comodules and free left contramodules over a given coalgebra are isomorphic.

To compare, when L:C->C is a monad and R:C->C is right adjoint to L, then R is a comonad and the whole categories of L-algebras and R-coalgebras in C are isomorphic.