Suppose $j:V\_\lambda \rightarrow V\_\eta$ is (elementary and) cofinal. Can $j$ be extended to all of $V$?


(Subsidiary question: What conditions are there on an ultrafilter/extender/whatever so that the induced $j:V \rightarrow M$ has $V\_\eta$ as an initial segment of $M$? Note that $j(\kappa)$ might well be less than $\eta$, where $\kappa$ is the critical point.)



extending elementary embeddings from initial segments of V to all of V
At successor ordinals, the answer is no, not necessarily, assuming the consistency of a nontrivial instance of your hypothesis. 


For a counterexample, let $\kappa$ be the least $1$-extendible cardinal. So there is $j:V\_{\kappa+1}\to V\_{\eta+1}$, and this map is elementary and cofinal, mapping $\kappa$ to $j(\kappa)=\eta$. 


I claim that this map cannot extend to $j:V\to M$ for a transitive set $M$, with $V\_{\eta+1}\subset M$. The reason is that the size of $j\upharpoonright V\_{\kappa+1}$ is $|V\_{\kappa+1}|=2^\kappa$, which is therefore coded by a size $2^\kappa$ subset of $V\_{\eta+1}$, which is therefore an element of $V\_{\eta+1}$. So $M$ will see that $\kappa$ is $1$-extendible, and so by elementarity there must be a $1$-extendible cardinal below $\kappa$, contradicting minimality. 


Similarly, if $\kappa$ is the least $(\theta+1)$-extendible cardinal, then there is $j:V\_{\kappa+\theta+1}\to V\_{\eta+1}$. But since $j\upharpoonright V\_{\kappa+\theta+1}$ is a size $|V\_{\kappa+\theta+1}|$ subset of $V\_{\eta+1}$, it is coded by an element of $V\_{\eta+1}$, and so if $j:V\to M$ extends $j$, then $M$ would see that $\kappa$ is $(\theta+1)$-extendible, violating minimality.


**Update.** But at limit ordinals $\lambda$, the answer is yes, all such elementary cofinal embeddings $j:V\_\lambda\to V\_\eta$ lift to $j:V\to M$ for some transitive class with $V\_\eta\subset M$. To see this, for any $a\in V\_\eta$, let $\mu\_a$ be the measure generated via $j$ by $a$, so that $X\in \mu\_a\iff a\in j(X)$. This measure will concentrate on any set $D\_a\in V\_\lambda$ for which $a\in j(D\_a)$, and such a set exists since the map was cofinal. Let $j\_a$ be the ultrapower of $V$ by $\mu\_a$. Elementary seed theory shows that $j\_a\upharpoonright V\_\lambda$ is a factor embedding of $j$, and furthermore, the range of $j\_a\upharpoonright V\_\lambda$ is precisely the seed hull $X\_a=\{\ j(f)(a)\mid f:D\_a\to V\_\lambda,\ f\in V\_\lambda\ \}\prec V\_\eta$. These elementary substructures form a directed system, and the map $j$ is the direct limit of the maps $j\_a\upharpoonright V\_\lambda$. But now the point is that the maps $j\_a$ are defined on all of $V$, since they are simply ultrapower maps by the measures $\mu\_a$. And so one may take the corresponding direct limit of the full maps $j\_a:V\to M\_a=\text{Ult}(V,\mu\_a)$. The result is an elementary embedding $j:V\to M$ into a transitive class (after the collapse), which extends the given map $j:V\_\lambda\to V\_\eta$. 


Basically, what is happening here is that we use the original map $j:V\_\lambda\to V\_\eta$ to define an extender, which is then applied to the whole of $V$. This strategy didn't work fully at succcessor ordinals $j:V\_{\theta+1}\to V\_{\eta+1}$, since there was no way to ensure that all of $V\_{\eta+1}$ was picked up in the range of the induced extender, as those seeds $a$ are on the top level of $V\_{\eta+1}$ and not covered by any element of $V\_{\theta+1}$ (and perhaps this could be considered a violation of cofinality of $j$, even though $j(\theta)=\eta$). But even in the successor ordinal case, we can still define the induced extender, and the resulting $j:V\to M$ will have $V\_\eta\subset M$ and it will agree with the original $j:V\_{\theta+1}\to V\_{\eta+1}$, and in particular include the image of $V\_{\theta+1}$ under $j$, although it may not have all of $V\_{\eta+1}$ contained in $M$. 


