Yes, you're right. It should be: newnode->next->prev = newnode; instead of newnode->next->prev = node_to_insert_after;

Aside from this, I'm not sure whether this is a typo but where you should have a }, you have {. For example, your struct definition should be:

struct node {
    int value;
    struct node* next;
    struct node* prev;
};

What I must ask is: why are you asking this question? Why not write some tests for yourself to see if things work rather than asking on stack overflow? For example, you could write a function which prints the list:

void print_list(struct node *head) {
    for(struct node *cur = head; cur != NULL; cur = cur->next) {
        printf("%d ", cur->value);
    }
    printf("\n");
}

Then you could use your insert functions and print the list to see if it works.