Cowedge under a Functor

A cowedge of a functor $F:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\mathcal{D}$ is an object $W$ along with a family of maps $w_{c}:F(c,c)\to W$ that satisfy the following Extranaturality condition:

• For all $f:\mathcal{C}(c,c^{\prime})$, the following diagram commutes \[

  \begin{tikzcd}

{F(c',c)} && {F(c', c')}

{F(c,c)} && W \end{tikzcd} \]