Question: what are $\lambda:1\times M\to M$ and $\rho:M\times 1\to M$? My guess is they are projections from the product to its right and left factor respectively. Is it correct? Moreover, is it possible to prove that they are isomorphisms?
EDIT: Let $\sigma:M\to 1$ be the unique morphism to $1$. Then $\tau:=\langle \sigma, 1_M\rangle:M\to 1\times M$ is such that $\lambda\tau=1_M$. This provides a right inverse to $\lambda$.
The definition given here is an instantiation of the more general concept of monoid in a monoidal category, instantiated in a cartesian (monoidal) category. Hence, the $\lambda$ and $\rho$ are the right and left unitors respectively, which correspond to the two projections in a cartesian category, as you indicate. It is possible to prove that projections $X \times 1 \to X$ and $1 \times X \to X$ are isomorphisms using the universal properties of the terminal object and cartesian product. I think this is a helpful exercise, but can provide hints if you're not sure where to start.