In the context of projective spaces, a homography is the most general transform. In $\mathbb{P}^2$ (projective space of $\mathbb{R^2}$), a homography can be represented as a matrix $H \in \mathbb{R}^{3 \times 3}$, which, nevertheless, has only 8 degrees of freedom, because, intuitively, in homogeneous coordinates, points are equivalent up to a scalar multiplication. In $\mathbb{P}^2$, both points and lines can be represented as vectors, so a homography can transform both points and lines.
In the context of computer vision specifically, why do we say that homographies transform points contravariantly and lines covariantly? What does that mean?
I have the following two equations
$$ m = (H^{-1})^{T} l $$
$$ \mathbf{x}' = H \mathbf{x} $$
where $H$ is a homography matrix, $m$ and $l$ are lines and $\mathbf{x}$ and $\mathbf{x}'$ are points in $\mathbb{P}^2$. Could this be the meaning of the expressions above? If yes, why exactly those expressions?
Define a function $f:\mathbb P^2 \rightarrow \mathbb P^2$ by $$f(x:y:z)=(x':y':z')$$ where $$\begin{bmatrix}x'\\y'\\z'\end{bmatrix}=H\begin{bmatrix}x\\y\\z\end{bmatrix}$$ Let $L$ be the line $\{(a:b:c)|la+mb+nc=0\}.$ The point $(x:y:z)$ is in $f(L)$ iff $$\begin{bmatrix}x\\y \\z\end{bmatrix}=H\begin{bmatrix}a\\b\\c\end{bmatrix}$$ for some $(a:b:c) \in L.$ $$\begin{bmatrix}l&m&n\end{bmatrix}\begin{bmatrix}a\\b\\c\end{bmatrix}=[0]$$ $$\begin{bmatrix}l&m&n\end{bmatrix}H^{-1}\begin{bmatrix}x\\y\\z\end{bmatrix}=[0]$$ Thus $f(L)$ is the line with equation $l'x+m'y+n'z=0$ where $$\begin{bmatrix}l'&m'&n'\end{bmatrix}=\begin{bmatrix}l&m&n\end{bmatrix}H^{-1}$$ $$\begin{bmatrix}l'\\m'\\n'\end{bmatrix}=(H^{-1})^T\begin{bmatrix}l\\m\\n\end{bmatrix}$$