Let $Y$ be path connected then every constant function $X\to Y$ are homotopic to each other.
I think I get what's the point of the question. I'm having some issues formally writing everything. The thing is that let $f,g: X\to Y$ two constant function, $f(x) = y_0, g(x) = y_1\; \forall x\in X$. Since $Y$ is path connected for every choice of $y_0,y_1$ there's a path $\alpha$ connecting this two points in $Y$. How do I formally write the homotopy? I can say that $F:X\times [0,1] \to Y $ that realizes the homotopy is $$F(x,t) = \alpha(t)$$
I think this is wrong or at least written very bad. Thank you for help.