$\newcommand{\cat}{\mathbf}\newcommand{\op}{\mathrm{op}}\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\cSet}{\cat{Set}}$A category $\cat C$ is total if the Yoneda embedding $\cat C→[\cat C^{\op},\cSet]$ has a left adjoint. This left adjoint sends $F ∈ [\cat C^\op,\cSet]$ to the coend $∫^x x·F(x)$. This is a strengthening of the notion of cocompleteness, where we require that diagrams indexed by large categories, but still controlled by the size of $\cat C$, have a colimit.
Let us call a functor $G : \cat C → \cat D$ totally cocontinuous (nonstandard terminology) if it preserves all these coends, meaning that $G(∫^x x·F(x)) = ∫^x G(x)·F(x)$ for all $F ∈ [\cat C^\op,\cSet]$. One can show that $G$ is totally cocontinuous if and only if it has a right adjoint (without condition on $\cat D$), which is $d ↦ ∫^x x·\Hom(G(x),d)$. I think it is the most direct analogue of the posetal case. So "totally cocontinuous functors" are just left adjoints functors, which probably explains why there is not another name for these.
One technically "has to be careful" because I imagine not every small cocontinuous functor (only colimits indexed by small categories are preserved) is totally cocontinuous. Can someone give me an idea of how bad things can go if we only assume small cocontinuity?
I found an example here (last paragraph), but the reason there is no right adjoint is that the target category is not locally small instead of the functor being small cocontinuous and not totally cocontinuous (the presheaf $\Hom(G(—),d)$ is not valued in sets). Is there an example of small cocontinuous functor $\cat C→\cat D$ with no right adjoint where $\cat C$ and $\cat D$ are locally small, and $\cat C$ is total?