In $\mathsf{Rel}$, are any two objects isomorphic? My knowledge of categories is rather basic, and I was just trying to find out what isomorphisms are in $\mathsf{Rel}$ where objects are sets and morphisms are relations. As far as I got, an isomorphism in $\mathsf{Rel}$ is any binary relation which is both left-total and right-total (projections on either factor set is that set), hence any two objects are isomorphic e.g. by a trivial relation. Am I missing something?
In general, if all the objects in some category are isomorphic, is there any interest in studying such category? The question arises for the reason that a lot of constructions in category theory (such as products) are defined up to isomorphism.
Edit: I did not put definitions of identity and composition in the OP as I thought that $\mathsf{Rel}$ is a pretty classical category -  at least google returns two top pages, Wikipedia and n-Lab, both providing necessary information. FWIW, by an identity I mean a diagonal of a set (equality relation) whereas the composition of two relations is as follows: for $R:X\to Y$ and $S:Y\to Z$
$$
  S\circ R = \{(x,z): \exists y\in Y \text{ s.t. }(x,y)\in R \text{ and }(y,z)\in S\}.
$$
I say that $R$ is left (respectively right) total whenever $\pi_X(R) = X$       (respectively $\pi_Y(R) = Y)$.
Going over the example proposed by @Amitai, I realized that my conditions imply $\mathrm{id}_X \subseteq R^{-1}\circ R$, but I've missed the fact that the RHS can be strictly bigger. Checking the isomorphism conditions thoroughly, I've found out that in $\mathsf{Rel}$ the isomorphisms are exactly graphs of bijections, hence isomorphic sets must be of the same cardinality - that was the guess I had in the beginning.
Nevertheless, I'm still interested in my second question above: is that of any interest to study a category where all objects are isomoprhic, besides of treating this as a monoid? For example, as far as I understood the category of groupoids has all morphisms being isomorphisms.
 A: It is not true that any left-total and right-total relation is an isomorphism. Consider $$A=\{0\},B=\{0,1\},R=\{(0,0),(0,1)\}.$$This $R$ does not have an inverse. In fact, these $A,B$ are not isomorphic.
A: To build a category, you need


*

*a class of objects

*for any two objects $A,B$ a set of morphisms $\operatorname{Mor}(A,B)$

*for any three objects $A,B,C$ a composition $\circ\colon \operatorname{Mor}(B,C)\times \operatorname{Mor}(A,B)\to \operatorname{Mor}(A,C)$

*for each object $A$ a specific morphism $\operatorname{Id}_A\in\operatorname{Mor}(A,A)$ (called identity)


such that the operation $\circ$ is associative where applicable and identity morphisms act as, well, identity where applicable.
An isomorphism $f\in\operatorname{Mor}(A,B)$ is then defined as a morphism for  which there exists $g\in\operatorname{Mor}(B,A)$ such that $f\circ g=\operatorname{Id}_B$ and  $g\circ f=\operatorname{Id}_A$.
After this recapitulization of the basic definitions, check again for $\mathsf{Rel}$: Objects are sets, mrphisms are relations. But what is $\circ$? What is $\operatorname{Id}_A$? What does it mean for a relation to be an isomorphism?

Regarding the question from your second paragraph: Even the study of one-object-only categories is of interest, and such a categroy can be identified with an algebraic structiure of a certain kind. Do you recognize it? 
