I have seen somewhere (emis.matem.unam.mx/journals/CMUC/pdf/cmuc9201/husek.pdf) that if $X$ has a reflection $r:X\longrightarrow Y$ in the class of compact spaces, then $r$ is an embedding and $Y$ may be found as a compactification of $X$.
Reflection: A space $Y$ from a class $C$ is a reflection of $X$ if $r:X\longrightarrow Y$ is a continuous map such that for every continuous mapping $f:X\longrightarrow Z$, where $Z$ is in $C$, there exists a unique continuous mapping $g:Y\longrightarrow Z$ such that $g \circ r=f$.
I have a problem with showing that $r$ is an embedding. Can it also happen that $r(X)$ is a dense subspace of $Y$?
Whether $r$ is an embedding depends on the class $\mathfrak{C}$. Consider for example the class $\mathfrak{H}$ of Hausdorff spaces. Then each $X$ has an $\mathfrak{H}$-reflection $r : X \to h(X)$ (see for example Does Hausdorffication preserve finite limits? and https://www.math.leidenuniv.nl/scripties/BachVanMunster.pdf). $r$ is a quotient map, in particular surjective. However, if $X$ is not Hausdorff, then $r$ is not injective, hence not an embedding. In fact, $r$ is an embedding if and only if it is a homeomorphism if and only if $X$ is Hausdorff.
Each Tychonoff space $X$ (also called $T_{3\frac{1}{2}}$-space) has reflection in the class $\mathfrak{CH}$ of comapct Hausdorff spaces. This is known as the Stone-Cech-compactification $\iota : X \to \beta X$ of $X$. See any book on general topology. In that case $\iota$ is an embedding, and $\iota(X)$ is dense in $\beta X$.