THis is a resume from my old
notes, the proofs aren't so difficult, but I include proof's if required....

PREMISES

Let $(F, G, \varepsilon , \eta): \mathscr{A} \to \mathscr{B}$ and adjunction.

Let $\Phi:{A, X}: (F(A), X)\cong (A, G(X)$ the natural bijection

give $f: F(A)\to X$ let $f^a:=G(f)\circ \eta_A$ its right adjoint
give $g: A\to G(X)$ let ${}^ag:=\epsilon_X\circ F(f)$ its left adjoint

For $f: A\to A'$ da ${}^a(\eta_A'\circ f)=\epsilon_{ F(A')} \circ F(\eta_{ A'})\circ F(f)= F(f)$ follow that

$F_{ A, A'} = \Phi_{ A, FA'}^{-1} \circ \mathscr{A}(A, \eta_{ A'}): \mathscr{A} (A, A') \to \mathscr{A} (A, G(F(A'))) \cong \mathscr{B}(F(A), F(A'))$

THEN WE HAVE THE FOLLOWING PROPERTIES:

**a)**
Give $G: \mathscr{C}\to \mathscr{A}$ let $\mathscr{A'} \subset\mathscr{A}$ the full subcategory with objects the $A\in \mathscr{A}$ such that $h^{A}_{G}: \mathscr{B}\to Set: B\mapsto (A, G(B))$ is representable

This is the maximum sub-category of which is defined a *partial left adjoint* $F$ of $G$, i.e. exist a bijection $\mathscr{C}(F(A), X)\cong \mathscr{A}(A, G(X))$ natural for $A\in \mathscr{A'}$ and $X\in \mathscr{B}$, then $F$ è unique but isomorphisms. Then $F$ preserves all colimits preserved by $\mathscr{A'} \subset_{fu}\mathscr{A}$ (also large or empty):

give a colimit cocone $(A_i\to A)_{i\in I} A_i$ in $\mathscr{A'}$ and a cocone
$e_i: (F(A_i)\to X)_{i\in I}$ from the cocone $(e_i^a : A_i \to G(X))_{i\in I}$ follow unique $g: A\to G(X)$ with $g\circ \epsilon_i=e_i^a$ then ${}^ag: F(A)\to X$ is such that ${}^ag\circ F(\epsilon_i)=e_i$, if $g', g'' : F(A)\to X$ verify the last condition then $g'^a, g''^a : A\to G(X)$ are equal, then $g'={}^a(g'^a)= {}^a(g''^a)=g''$. Is easy proof that $F$ preserving epimorphisms, and dually $G$ preserving monomorphisms, and $F$ preserving strong.epimorphisms and dually $G$ preserving strong-monomorphisms.

**b)**
The following properties are equivalent:

b.1) $F$ is faithful (full, full and faithful)

b.2) $\eta$ is a pointwise-monomorphism (pointwise-Retraction, a Isomorphism)

b.3) $F$ reflect monomorphism

b.4) $\Phi_{ A, B }$ preserving monomorphisms

b.5) For any $X\in\mathscr{C}$ the source $(a:X\to G(A))_{A\in \mathscr{A}, a\in (A, G(A))}$ is a mono-source (is enough considering $A$ belong to cogenerating class).

.

In Particular if $F$ is full from $1_G=G\varepsilon * \eta G$, $1_F= \varepsilon F*F\eta$ follow that $\eta G$, $G\varepsilon $, $F\eta$, $\varepsilon F$ are isomorphisms.

**c)**
Here we call $F$ conservative is reflect isomorphisms, and call a morphisms $m: A\to B$ a
co.cover if from $m=f\circ e$ with $e$ epimorphism follow that $e$ is a isomorphism, for straight generalization we have the definition of cocover source.

We have the implication:
(1) $F$ is conservative $\Rightarrow $ (2) $F$ reflect co.Cover's $\Rightarrow $ (3) $\eta$ is pointwise-co.cover $\Leftrightarrow$ The source $(a:X\to G(A))_{A\in \mathscr{A}}$ is a co.cover source.

And $(3)\Rightarrow(1)$ if $F$ reflect isomorphisms on epimorphisms (I.e. if $F(e)$ is a isomorphism then $e$ is a epimorphism, in particular this happen if $F$ is faithful).

**d)**
We call $F: \mathscr{B}\to \mathscr{A}$ co.fiathfull if for $H, K: \mathscr{A}\to \mathscr{C}$ and $\phi, \psi: H\to K$ and $\phi\circ F= \psi\circ F$ follow that $\phi=\psi$.
ANd call $F$ co.conservative if (on the data above) from $\phi\circ F$ isomorphisms follow that $\phi$ is isomorphism.

We have the following equivalent properties:

d.1) $G$ if full and faithful

d.2) $\epsilon$ is isomorphism

d.3) $F$ is dense

d.4) $F\circ U$ is dense for some (any) $U: \mathcal{C}\to \mathscr{A}$ dense

d.5) the functor $F^*: \mathscr{B}[\Sigma]\to \mathscr{A}[\Sigma]$

where $\Sigma:=F^{-1}(Iso)$ , $F=F^*\circ P$, and $P: \mathscr{B}\to \mathscr{B}[\Sigma]$ canonic, is a equivalence

d.6) $F$ is co.fauthful $\Rightarrow$ $F$ is co.conservative.

**e)** G riflect strong.epimorphisms $\Leftrightarrow$ $\epsilon$ is pointwise-strong.epimorphisms

**f)** If $G$ is full and $\eta$ is pointwise-Section then $\eta$ is a Isomorphism.

**g)** Define a epimorphisms $e: X\to Y$ a (small)source-strong-epimorphism if give $f: X\to A$ and a (small) monosource $(m_i: A\to A_i)_{i\in I}$ and a (small) source $(g_i: Y\to A_i)_{i\in I}$ with $g_i\circ e=m_i\circ f\ i\in I$ exist unique a diagonal $d: Y\to A$ that keep the commutativity of the diagram.

We have te following property:

If for any $A\in \mathscr{A}$ the morphism $\epsilon_A : FG(A)\to A $ is (small)source-strong-epimorphism then $G$ reflect large (small) limits.\

**h)** Let $F$ such that for $X\in \mathscr{C}$ we have $1_X=s\circ r: X\to F(A)\to X$ for some $s,\ r$.
From $\epsilon_X\circ FGF(r)=r\circ \varepsilon _{ FA }$ where $r$ and $\epsilon _{F(A)}$ retractions follow that $\epsilon_X$ is a retraction, then a epimorphisms and $G$ is faithful. If $G_{ A, A'}: \mathscr{B}(F(A), F(A'))\to \mathscr{A}(GF(A), GF(A'))$ is surjective then $G$ is full:

for $u: G(B_1)\to G(B_2)$ with $1=\rho_k\circ \sigma_k: A_k\to F(B_k)\to A_k$ follow $G(\sigma _2)\circ u\circ G(\rho_1): GF(B_1)\to GF(B_2)$ and this is $G(v)$ for some $v: F(B_1)\to F(B_2)$, then $u=G\sigma _2\circ v\circ Q\rho_1$.

**i)**

Give the adjoint couples $(U_! , U^\ast)$ and $(U^\ast, U_\ast)$ where

$U^\ast: \mathscr{A}\to \mathscr{E}$.

For a category $\mathscr{C}$ let
$\mathscr{C}^>:=Fun(\mathscr{C}^{op}, Set)$ the category of presheaves .We have the following equivalents properties:

i1) $U_!$ is faithfull and full (faithful).

i2) The unity $\eta_H: H\to U^\ast U_!(H)$, for $H\in \mathscr{A}^>$ is a isomorphisms (a monomorphism).

i3) $U_\ast$ is faithfull and full (faithful).

i4) The counity $\epsilon_H: U^\ast U_\ast (H)\to H$, for $H\in \mathscr{A}^>$ is a isomorphisms (a epimorphism).