Proving that binary products exists

Question

I have to show that $\textbf{Mon}$ the category of monoids have all finite products. For this we require the existence of a terminal object (a nullary product) and the existence of all binary products. Now the monoid with only one element $1$ can play the role of the terminal object (correct?). But to show that all binary products exists I have to show that all product diagrams have the UMP (universal mapping property), that is having a binary product $M\times N$ with projections $\pi_1,\pi_2$ being $\pi_1:M\times N\to M$ (I guess taking $(m,n)\mapsto (m,1)$) and similar for $\pi_2$, then if we have two other maps into $M$ and $N$ from an object $T$ ($\tau_1:T\to M$ and $\tau_2:T\to N$) we will get a so called mediating morphism $\alpha :T\to M\times N$ which fullfills $\tau_1=\pi_1\circ \alpha$ and the same for $\tau_2$. And here I am stuck. Any hints would be really appreciated.