# Definitions of free and based homotopy.

I have a question concerning the right understanding of the following two definitions: free and based homotopy.

If I am right, free homotopy between two continuous maps $f, g : X \to Y$ is a continuous map $F: X \times I \to Y$ such that

\begin{align*} F(x,0) &= f(x),\ \ \forall x\in X\\ F(x,1) &= g(x),\ \ \forall x \in X \end{align*}

(with no extra condition that these two maps have to be based maps?)

The other definition (based homotopy) would be the following:

For two based maps $f,g: (X, x_0) \to (Y, y_0)$ one says that they are based homotopic if there exists a continuous map $G : X \times I \to Y$ such that

\begin{align*} G(x,0) &= f(x),\ \ \forall x\in X\\ G(x,1) &= g(x),\ \ \forall x \in X\\ G(x_0,t) &= y_0,\ \ \forall t \in I \end{align*}

Does the second definition imply that we can speak only about based homotopy when having based maps?

You can talk about free homotopies between any two maps $f, g : X \to Y$. If $f$ and $g$ are based maps, you can still talk about free homotopies between them, you just ignore the fact that they are based maps (in particular, the basepoint plays no role).