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In this paper, we determine the complex-valued solutions of the functionalequation $$ f(x\sigma(y))+f(\tau(y)x)=2f(x)f(y)$$ for all $x,y \in M$, where$M$ is a monoid, $\sigma$: $M\longrightarrow M$ is an involutive automorphismand $\tau$: $M\longrightarrow M$ is an involutive anti-automorphism. Thesolutions are expressed in terms of multiplicative functions, and characters of$2$-dimensional irreducible representations of $M$.

A variant of d'Alembert functional equation on monoids