Equation in the paper "homotopy limits for 2-categories"

Question

I do not understand a step in the paper "Homotopy Limits for 2-Categories" by Nicola Gambino. The two longer calculations on page 22 together imply that \begin{align} \textrm{Ps}(\mathscr J,\textrm{Cat})(W, K(X,D-)) \cong \textrm{Ps}(\mathscr J,K)(W(-) \ast X,D) \end{align} when $K$ is a 2-category with copowers denoted by $\ast$. I do not see how the equation above follows from the universal property \begin{align} \textrm{Cat}(W,K(A,B)) \cong K(W\ast A,B) \end{align} of cotensors in $K$. Can somebody help me?

Answer 1

This is a very short computation with the help of pseudo-ends, whose theory is developed in the very recent text Notes on Lax Ends by Kengo Hirata. One can compute: \begin{align*} \operatorname{Ps}(\mathscr J,K)(W,K(A,D-)) &\cong \textstyle \int_X^{ps} \operatorname{Cat}(WX,K(A,DX)) \\ &\cong \textstyle \int _X^{ps}\operatorname{Cat}(WX*A,DX) \\ &\cong \operatorname{Ps}(\mathscr J,K)(W(-)*A,D) \end{align*}