It's fairly common to identify the tangent space of a product manifold as $$ T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N $$ where $p=(p_1,p_2)$, and the actual isomorphism is given by $v\in T_p(M\times N)$ is sent to $(d(\pi_M)_p(v),d(\pi_N)_p(v))$, where the $\pi_N$ and $\pi_M$ are the projections.
If we make this identification, how do differentials act on the identified elements? I mean, suppose you have a smooth map $F:M\times N\to S$, and a corresponding differential $dF_p:T_p(M\times N)\to T_{F(p)}S$.
If $v\in T_p(M\times N)$, using the isomorphism I could identify $v$ with $(v_1,v_2)$ for $v_1\in T_{p_1}M$, $v_2\in T_{p_2}N$. I'm confused on how one could evaluate $dF_p$ on $(v_1,v_2)$, since it's in a new tangent space, albeit an isomorphic one.
What would be the correct way to evaluate $dF_p(v_1,v_2)$ after this identification?