Given a category$C$ and an object$c \in C$, the under category (also called coslice category) $c \downarrow C$ (also written $c/C$ and sometimes, confusingly, $c\backslash C$) is the category whose

objects are morphisms in $C$ starting at $c$; $c \to d$

morphisms are commuting triangles $\array{
&& c
\\
& \swarrow && \searrow
\\
d_1 &&\to && d_2
}
\,.$

The under category $c\downarrow C$ is a kind of comma category; it is the strict pullback

$\array{
c\downarrow C
&\to&
pt
\\
\downarrow
&&
\;\;\downarrow^{pt \mapsto c}
\\
[I,C]
&\stackrel{d_0}{\to}&
C
}$