Let $C, C', C''$ be three categories. Let $C \xrightarrow{F\_1, F\_2} C'$ and $C' \xrightarrow{G\_1, G\_2} C''$, be four functors, and let $\theta : F\_1 \Rightarrow F\_2, \ \lambda : G\_1 \Rightarrow G\_2$, be two natural transformations. Then I've already shown algebraically that if we defined $(\lambda \* \theta)\_X \equiv G\_2(\theta\_X) \circ \lambda\_{F\_1 X}$, then $\lambda \* \theta$ is a natural transformation $G\_1 \circ F\_1 \Rightarrow G\_2 \circ F\_2$. Now I want to show that for all $X \in C$, the following diagram commutes (or the two formulas for $\lambda \* \theta$ are the same. A book says by naturality of $\theta$ and functoriality of $G\_1$ the square commutes. But I'm having trouble seeing this. Please explain using diagrams or give a hint.

[![enter image description here](https://i.stack.imgur.com/AHzi4.jpg)](https://i.stack.imgur.com/AHzi4.jpg)