I am currently studying the book Elements of $\infty$-Cats and stumbled across the following Corollary (this is Corollary 4.1.3 on page 88 in the book):
I am a bit confused on how to get the induced fibered equivalence $\text{Hom}_A(fb,a) \simeq \text{Hom}_B(b,ua)$ is obtained from the fibered equivalence $(\star) \text{Hom}_A(f,A) \simeq \text{Hom}_B(B,u)$. In principle, I know it would be enough to see that the map $(\star)$ plus the comma category $\text{Hom}_A(fb,a)$ induce a cone for the diagram underlying the limit $\text{Hom}_B(b,ua)$ - yet I am struggling to see this somehow.
Note: The comma $\infty$-categories above are defined by
In the commutative diagram $$\require{AMScd}\begin{CD} \mathsf{Hom}_A(fb,a) @>>> \mathsf{Hom}_A(f,A)@>{\varphi}>> A^{\mathbf{2}}\\ @VVV @VV{(p_1,p_0)}V @VV{(p_1,p_0)}V\\ X\times Y @>{a\times b}>> A\times B @>{\mathrm{id}_A\times f}>> A\times A \end{CD}$$ both the right square and the outer rectangle are pullback squares, and hence the left square is also a pullback square. Likewise, the square $$\require{AMScd}\begin{CD} \mathsf{Hom}_B(b,ua) @>>> \mathsf{Hom}_B(B,u)\\ @VVV @VV{(p_1,p_0)}V\\ X\times Y @>{a\times b}>> A\times B \end{CD}$$ is a pullback square. But then the equivalence $\mathsf{Hom}_A(f,A)\simeq\mathsf{Hom}_B(B,u)$ over $A\times B$ can be pulled back to an equivalence $\mathsf{Hom}_A(fb,a)\simeq\mathsf{Hom}_B(b,ua)$ over $X\times Y$ (as we are pulling back along an isofibration). You could also phrase this as pullbacks along isofibrations being stable under replacing an object in the cospan by an equivalent object.