Let $\mathbf{C}$ be a nice enough category. Given arrows $f:a\to b$ and $g:c\to d$ there exists a product arrows $$f\times g:a\times c\to b\times d$$ which is $f\times g=\langle f\pi_a,g\pi_c\rangle$ (where $\pi_a$ and $\pi_b$ are the usual projections). This is well known.
Now, let $a$ be an object of $\mathbf{C}$ and $\mathbf{C}/a$ be the usual slice category.
Every $\mathbf{C}/a$-arrow $f$ between two $\mathbf{C}/a$-objects $(x:X\to a)$ and $(y:Y\to a)$ can be "lifted" (forgetful functor) to a $\mathbf{C}$-arrow $X\to Y$, which I denote by $\lceil f\rceil$.
Also, some $\mathbf{C}$-arrows $g$ can be "dropped" to a $\mathbf{C}/a$-arrows which I denote by $\lfloor g\rfloor$. I hope this notation is clear.
Now, in the same situation, i.e given $\mathbf{C}/a$-arrows $$f:(r:\alpha\to a)\to(s:\beta\to a)$$ $$g:(t:\gamma\to a)\to (u:\delta\to a)$$ (it is implied that $s\lceil f\rceil=r$ and $u\lceil g\rceil=t$) there exists a product arrow $$f\times g:r\times t\to s\times u$$ which lifts to the $\mathbf{C}$-arrow $$\lceil f\times g\rceil:Pb\left(\begin{array}{l} r:\alpha\to a\\ t:\gamma\to a\end{array}\right)\to Pb\left(\begin{array}{l} s:\beta\to a\\ u:\delta\to a\end{array}\right)$$
where $$Pb\left(\begin{array}{l} r:\alpha\to a\\ t:\gamma\to a\end{array}\right)$$ denotes the pullback of the arrows $r$ and $t$ and it is the domain of the $\mathbf{C}/a$-object $(r:\alpha\to a)\times(t:\gamma\to a)$.
(Products in $\mathbf{C}/a$ correspond to pullbacks in $\mathbf{C}$ and vice versa)
The question is how, if possible, can we put $\lceil f\times g\rceil$ in terms of $\lceil f\rceil$ and $\lceil g\rceil$ ?
The dream of $\lceil f\times g\rceil=\lceil f\rceil\times\lceil g\rceil$ isn't true because the domain and codomain of $\lceil f\rceil\times\lceil g\rceil$ are $\alpha\times\gamma$ and $\beta\times\delta$ respectively (both different from the (co)domain of $\lceil f\times g\rceil$)
Can you find a nice expression? It may involve other arrows like $\pi_\beta$, $\pi_\alpha$, $\lceil\pi_t\rceil$, ...
Thanks!