  The coupling of matter to supergravity with $N=1$ supersymmetry in $d=4$
dimensions is described in a geometric manner by K\"ahler superspace. A
straightforward way to implement K\"ahler superspace is via $\mathrm{U}(1)$
superspace by identifying the $\mathrm{U}(1)$ pre-potential with the K\"ahler
potential, which is a function of the matter (chiral) superfields. In this
framework, the components of the supergravity multiplet are contained in the
supervielbein and torsion tensor of superspace. Furthermore, interactions with
the Yang-Mills (vector) multiplet are formulated by introducing a connection
$1$-superform of an additional gauge structure. In these notes, the Bianchi
identities in $\mathrm{U}(1)$ superspace are solved for a particular set of
torsion constraints which lead to the minimal supergravity multiplet. Moreover,
the solution of the Bianchi identities in the gauge sector is derived and
K\"ahler superspace is defined. At the superfield level, the general action of
the supergravity/matter/Yang-Mills system and supergravity transformations are
formulated, and the equations of motion are deduced. Using projection to lowest
components in superspace, the corresponding Lagrangian and supergravity
transformations at the component field level are calculated, and the equations
of motion of the auxiliary fields are determined. Compared to existing
literature, these notes provide a self-contained and consistent step by step
derivation of the general supergravity/matter/Yang-Mills Lagrangian for $N=1$
supersymmetry in $d=4$ dimensions by means of superspace techniques.
