h = yonedaProduct(f,g)
Given a triple $(L, M, N)$ of $R$-modules, the Yoneda product is a pairing between $\operatorname{Ext}$-modules
$\phantom{WWWW} \operatorname{Ext}_R^d(L,M) \otimes \operatorname{Ext}_R^e(M,N) \to \operatorname{Ext}_R^{d+e}(L,N). $
For an element of $\operatorname{Ext}_R^{e}(M,N)$, thought of as an extension
$\phantom{WWWW} 0 \leftarrow M \leftarrow F_{0} \leftarrow F_{1} \leftarrow \dotsb \leftarrow F_{e-2} \leftarrow P \leftarrow N \leftarrow 0, $
and for an element of $\operatorname{Ext}_R^{d}(L,M)$, thought of as an extension
$\phantom{WWWW} 0 \leftarrow L \leftarrow G_{0} \leftarrow G_1 \leftarrow \dotsb \leftarrow G_{d-2} \leftarrow Q \leftarrow M \leftarrow 0, $
the Yoneda product corresponds to
$\phantom{WWWW} 0 \leftarrow L \leftarrow G_{0} \leftarrow G_{1} \leftarrow \dotsb \leftarrow Q \leftarrow F_{0} \leftarrow F_{1} \leftarrow \dotsb \leftarrow P \leftarrow N \leftarrow 0, $
where the map from $F_0$ to $Q$ factors through $M$. For more information about extensions, see yonedaExtension.
Alternatively, the module $\operatorname{Ext}^d_R(L,M)$ is constructed from a free resolution $G$ of $L$,
$\phantom{WWWW} 0 \leftarrow L \leftarrow G_0 \leftarrow G_1 \leftarrow \dotsb \leftarrow G_d \leftarrow \dotsb, $
by taking the homology of the complex $\operatorname{Hom}_R(G, M)$. An element of $\operatorname{Ext}^d_R(L,M)$ is represented by an element of $\operatorname{Hom}_R(G_d, M)$. This map extends to a complex map having degree $-d$ from $G$ to the free resolution $F$ of $M$. The Yoneda product is the composition of the map of chain complexes from $G$ to $F$ with the map of chain complexes having degree $-e$ from $F$ to a free resolution of $N$. For more information about these maps, see yonedaMap.
As an example, we take two distinct elements of an $\operatorname{Ext}^1$-module to obtain a non-zero element of the $\operatorname{Ext}^2$-module.
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In our second example, all three modules in the triple are distinct and the image of the Yoneda product is not surjective.
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The source of this document is in Complexes/ChainComplexDoc.m2:4040:0.