If a category has pullbacks and a terminal object, then it has (binary) products

$\require{AMScd}$ The questions is in the title.

Here is what I have attempted. I want to know if this is enough.

Let $1$ be the terminal object in a category $K$. Given objects $X, Y$ consider the pullback diagram:

$$\begin{CD} P @>p_{1}>> X \\ @Vp_{2}VV @VVV\\ Y @>>>1\\ \end{CD}$$

therefore we have the product cone:

$$\begin{CD} P @>p_{1}>> X \\ @Vp_{2}VV \\ Y \end{CD}$$

Consider the product cone :

$$\begin{CD} A\times B @>\pi_{A}>> A\\ @V\pi_{B}VV \\ B \end{CD}$$

$$\begin{CD} X @>f>>A\\ @V g VV\\ B \end{CD}$$

then $$X \xrightarrow[g]{f} A \times B$$

is the unique factorization through the product. is that enough?

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You gave the idea, but you have of course to show that the pullback is a product. That is given an object $A$ and morphisms $\xi\colon A \to X$ and $\eta\colon A \to Y$ that there is a unique morphism $(\xi,\eta) \colon A \to P$ such that $\xi = p_1 \circ (\xi,\eta)$ and $\eta = p_2 \circ (\xi, \eta)$. For doing so you can use the properties of the pullback ...