Gallery – Profunctors

(Note that we write compositions from left to right and accordingly apply most maps on the right.)

Def.1
A functor $F:\ct A\to\ct B$ is called a left adjoint to a functor $U:\ct B\to\ct A$ if there is a bijection between the homsets:

$\ct B(A^F\ig B) \simeq \ct A(A\ig B^U)$

natural in both $A$ and $B$ . In this case $U$ is called a right adjoint to $F$ , and we write $F\adj U$ .

Def.2
A functor $F:\ct A\to\ct B$ is called a left adjoint to a functor $U:\ct B\to\ct A$ if there are natural transformations $\eta:1_{\ct A}\tto FU$ and $\eps:UF\tto 1_{\ct B}$ satisfying the zig-zag identities:

$\matrix{\eta F\cdot F\eps = 1_F &\sep U\eta\cdot \eps U=1_U}$

So, why are these two definitions equivalent?
Continue reading “On the two definitions of adjunctions”