Current flowing through a battery
Making the current flow the wrong way through a battery
The workings of a battery are pretty complex and it's very easy to get confused if you try and think too hard how charges can flow through one. If you're happy with the idea of conventional current then you can probably convince yourself that the flow of positive and negative ions in a battery can be thought of as a current flowing straight through it.
A good demonstration is to make the current flow backwards through a single battery. This should convince you that there must be a conducting path through the battery rather than it having two unconnected ends.