We know that the algebra and coalgebra axioms are given via following commutative diagrams (algebra $A$ and coalgebra $C$ are over a field $\mathbb{K}$):
I am now trying to show that the dual of a coalgebra $C^*$ has an algebra structure given by $$\mu_{C^*} := \Delta^* \circ \lambda \quad \text{and} \quad \eta_{C^*} := \varepsilon^*,$$ where $$\begin{align*}\lambda : C^* \otimes C^* &\longrightarrow (C \otimes C)^* \\ p \otimes q &\longmapsto [x \otimes y \longmapsto p(x)q(y)].\end{align*}$$
My question is the following: I can verify the axioms by a painful direct calculation, but the book I am using simply says that the result follows from considering the diagrams above. How do I actually consider these diagrams? From my understanding, they simply encode the axioms in a neat way (e.g. we do not have to write out explicitly what the coproduct does to elements of $C$), but I do not see how from simply looking at the diagrams we can deduce the result in question.