Passing pullbacks through adjunction

Question

I'm having trouble following the proof of Proposition I.5.2 in Goerss-Jardine (Simplicial Homotopy Theory). After establishing the adjunction $\hat\Delta(X\times K,Y) \simeq \hat\Delta(K,[X,Y])$, they claim that diagrams of the form

correspond via the exponential adjunction to diagrams of the form

where $i\colon K\hookrightarrow L$ is an inclusion, and $p\colon X\to Y$ is a fibration, and the pullback is induced by

I imagine this is at the level of trivial manipulations with the adjunction, but I can't get it to come out right (mainly not sure how to go from the pullback to the pushout).

Answer 1

Apart from performing a somewhat lengthy diagram chase, the only thing to do here is to notice that giving a map $(a,b) : A \to [K,X] \mathrel{\times_{[K,Y]}} [L,Y]$ amounts to giving a commutative diagram as on the left by the universal property of the pull-back $\DeclareMathOperator{\Hom}{Hom}$

while the diagram on the right is equivalent to it by the adjunction between the cartesian product and the internal $\Hom$ and the definitions of $p_\ast$ and $i^\ast$.

Let me give names to the maps in the first commutative diagram of the proof you ask about:

The map $f: \Lambda_{k}^n \to [L,X]$ corresponds to a map $\tilde{f}: \Lambda_{k}^n \times L \to X$ while giving the bottom map $(g,h) : \Delta^n \to [K,X] \mathrel{\times_{[K,Y]}} [L,Y]$ amounts to either of the two commutative diagrams

Using the universal property of the pull-back defining $[K,X] \mathrel{\times_{[K,Y]}} [L,Y]$ we see that asserting the commutativity of the square we started with is equivalent to giving the map $f: \Lambda_{k}^n \to [L,X]$, the commutative square $(1)$ and requiring the two squares

to be commutative.

Passing to the adjoint side using the map $\tilde{f} : \Lambda_{k}^n \times L \to X$ and the square $\widetilde{(1)}$, the commutativity of the squares $(2)$ and $(3)$ is equivalent to the two commutative squares

At this point I think I can leave it to you to contemplate the commutative diagrams $\widetilde{(1)}$, $\widetilde{(2)}$ and $\widetilde{(3)}$ and the push-out square defining $(\Lambda_{k}^n \times L) \mathrel{\cup_{(\Lambda_{k}^n \times K)}} (\Delta^n \times K)$ and to think about what it means to define a map from $(\Lambda_{k}^n \times L) \mathrel{\cup_{(\Lambda_{k}^n \times K)}} (\Delta^n \times K)$ in order to see that giving the squares $\widetilde{(1)}$, $\widetilde{(2)}$ and $\widetilde{(3)}$ and requiring their commutativity is equivalent to giving the commutative square

as claimed by Goerss and Jardine.

Answer 2

Halfway through typing my answer I saw that t.b. already gave it. Since I seem to be unable to comment on anything, I post this comment as an answer. The original problem is certainly not trivial because there is more than one pair of adjoint functors: you have $(-)\times K \dashv [K,-]$ and $(-)\times L\dashv [L,-]$.

The missing concept is that of conjugate natural transformations as explained in Mac Lane's CWM (Section IV-7): suppose $F,F':\mathcal{X}\rightarrow\mathcal{A}$ and $G,G':\mathcal{A}\rightarrow\mathcal{X}$ are functors with $F\dashv G$ and $F'\dashv G'$. Two natural maps $\alpha: F\rightarrow F'$ and $\beta: G'\rightarrow G$ (note the opposite directions) are said to be conjugate if the diagrams $$ \matrix{ & \mathcal{A}(F'x,a) & \cong & \mathcal{X}(x,G'a) & \cr \mathcal{A}(\alpha_x,a) &\Big\downarrow& &\Big\downarrow& \mathcal{X}(x,\beta_a)\cr & \mathcal{A}(Fx,a) & \cong & \mathcal{X}(x,Ga) & } $$ commute for all objects $x\in\mathcal{X}$, $a\in\mathcal{A}$, where the horizontal maps are the isomorphisms given by adjointness. In fact, any given $\alpha: F\rightarrow F'$ determines a unique conjugate $\beta=\alpha^*: G'\rightarrow G$ (take $x=G'a$ and start with $id_{G'a}$ in the upper right corner).

In your situation the inclusion map $i:K\hookrightarrow L$ gives a natural map $(-)\times i:(-)\times K\rightarrow (-)\times L$ and the corresponding conjugate is $i^*:[L,-]\rightarrow [K,-]$ given by precomposing with $i$. This is conveniently already named $i^*$ in your diagram, although with objects dropped from the notation.

I always wondered why this notion of conjugate transformations is never made explicit in books on algebraic topology although it is used all the time.