Function is to relation as functor is to?

I am told that the answer to the question in the title is a profunctor. However, below I sketch a different way that one could think about structure-preserving relations between categories. My questions are

1. Is the thing I define below the same thing as a profunctor (for small categories at least)? If so, how can I see the connection?

2. If not, does the thing I define have a name, and can it be satisfactorily defined in general, not just for small categories? And is it useful for anything?

Here's my definition:

Let $C$ and $D$ be small categories. Define a relator (for want of a better name) as a relation $\sim$ between ${\rm Ob}(C)$ and ${\rm Ob}(D)$ and a relation that I'll also denote $\sim$ between the morphisms of $C$ and the morphisms of $D,$ such that:

• For $P\in {\rm Ob}(C)$ and $X\in {\rm Ob}(D)$, if $P\sim X$ then ${\rm Id}_P \sim {\rm Id}_X$
• For morphisms $f:P\to Q$ in $C$ and $u:X\to Y$ in $D$, if $f\sim u$ then $P\sim X$ and $Q\sim Y$.
• For morphisms $f:P\to Q$ and $g:Q\to R$ in $C$, and $u:X\to Y$ and $v:Y\to Z$ in $D$, if $f\sim u$ and $g\sim v$, then $f;g \sim u;v$.

The point being that a functor is just a special case of this, in which every object in $C$ relates to exactly one object in $D$, and every morphism in $C$ relates to exactly one morphism in $D.$

I restricted it to small categories because a relation is a subset of the Cartesian product, which isn't defined for proper classes. I do not have a good idea about how it would generalise.

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