= Solution
Let $p_i:M_1\times M_2\to M_i$ be the projections. The derivative of the two projections identifies the <tangent bundle> with
$$
T(M_1\times M_2)\cong p_1^*TM_1\oplus p_2^*TM_2.
$$
Take the <direct sum> of the two <pullback connections>. This is the <product affine connection>. In product coordinates $(x^a,y^\alpha)$, its only nonzero <Christoffel symbols> are the two blocks inherited from the factors; all mixed coefficients are zero.
To make the connection on arbitrary fields explicit, write $V=V^a\partial_{x^a}+V^\alpha\partial_{y^\alpha}$ and $W=W^a\partial_{x^a}+W^\alpha\partial_{y^\alpha}$, allowing all coefficients to depend on both variables. Then
$$
\boxed{\begin{aligned}
(\nabla_VW)^a&=V(W^a)+(\Gamma^1)^a{}_{bc}(x)V^bW^c,\\
(\nabla_VW)^\alpha&=V(W^\alpha)+(\Gamma^2)^\alpha{}_{\beta\gamma}(y)V^\beta W^\gamma.
\end{aligned}}
$$
The terms $V(W^a)$ and $V(W^\alpha)$ include derivatives in both factors. Omitting these cross derivatives for fields with variable coefficients would not define a connection.
The formula is real-bilinear, linear over <smooth functions> in $V$, and obeys the <Leibniz rule> in $W$. Under a change of product coordinates each block has the factor's connection transformation law, so the local formulas agree; equivalently the <pullback connection> construction already guarantees this gluing.
A field $X_i$ on one factor has a canonical lift to the product. Opposite-factor lifts commute, and their mixed covariant derivatives vanish. For such lifted fields the coefficient derivatives reduce to their factor derivatives, giving
$$
\boxed{\nabla_{Y_1+Y_2}(X_1+X_2)=\nabla^1_{Y_1}X_1+\nabla^2_{Y_2}X_2.}
$$
These values determine the connection uniquely: lifted coordinate frames span locally, and the connection's product rules determine its action on all their smooth linear combinations.
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